Math 1111 Calculus (I)
Homework 11
1. Compute (a) lim
x→0
cosx−cos 3x sin(x2) (b) lim
x→1
2
x2 −1 − 1 x−1
(c) Iff0 is continuous,f(2) = 0, and f0(2) = 7, evaluate
x→0lim
f(2 + 3x) +f(2 + 5x) x
(d) For what values ofa and b is the following equation true?
x→0lim
sin 2x
x3 +a+ b x2
= 0 2. (87’ Calculus Exam)
(a) Suppose that f(x) is continuous on [a, b] and differentiable on (a, b). Prove that if lim
x→c+f0(x) = A, then f0(c) = A for some c∈(a, b).
(b) Suppose thatf :R→Rwith f(0) = 0 and |f| is differentiable at 0. Prove that f is differentiable at 0 and f0(0) = 0.
(c) Letf(x) =
( 1−cosx
x , x6= 0
0, x= 0
. Prove that f is a differentiable function and f0 is continuous at 0.
3. (90’ Calculus Exam) Discuss the increase, decrease, concavity, ex- treme values and inflection point(s) of the function
f(x) = (x−1)13 −2(x−1)43 and sketch it graph.
4. (91’ Calculus Exam) Suppose that f is differentiable on R with f(0) = 0 and g(x) = f(x)
x for all x > 0. Prove that if f0 is increasing on [0,∞), then g is increasing on (0,∞).
5. (92’ Calculus Exam) Let f(x) be a 3 degree polynomial function with 3 distinct roots and a be the average of these three roots. Prove that (a, f(a)) must be an inflection point.
6. Use the guidelines we discuss in class to sketch the curve (a) y= sinx
2 + cosx (b) y=x+ cosx
7. Let f : (a, b) → R be a continuous and 1-1 function. Prove that the range of f is an open interval.
8. Find the inverse functions of the given functions and find their domains and ranges
(a) f(x) = 3x−5 2x+ 1
(b) f(x) = x3−3x2+ 3x−1 (c) f(x) = tan−1(x3)
